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Full text of "Men Of Mathematics"

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THE GREAT ALGORIST that Jacob! made Ms great start in entire ignorance of bis rival's work. A capital property of the elliptic functions is their double periodicity (discovered in 1825 by Abel): if E(x) is an elliptic function, then there are two distinct numbers, say pv p2, such that E(x + pi) = E(x), and Efa for all values of the variable #, Finally, on the historical side, is the somewhat tragic part played by Legendre. For forty years he had slaved over elliptic integrals (not elliptic functions) without noticing what both Abel and Jacobi saw almost at once, namely that by inverting his point of view the whole subject would become mfinitely simpler. Elliptic integrals first present themselves in the pro- blem of finding the length of an arc of an ellipse. To what was said about inversion in connexion with Abel the following statement in symbols may be added. This will bring out more clearly the point which Legendre missed, If R(t ) denotes a polynomial in t , an integral of the type 1 r J ( is called an elliptic integral if E(t) is of either the third or the fourth degree; if R(t) is of degree higher than the fourths the integral is called Abelian (after Abel, some of whose greatest work concerned such integrals). If R(t) is of only the second degree, the integral can be calculated out in terms of elementary functions. In particular f* 1 — dt J o Vl - (sin-% is read, 'an angle whose sine is a?*). That is, if we consider the upper limit, a, of the integral, as a function of the integral itself, namely of y. This inversion of the problem 369